Lemma 88.10.1. Let $A$ be a Noetherian ring and $I = (a)$ a principal ideal. Let $B$ be an object of (88.2.0.2) which is rig-étale over $(A, I)$. Then there exists a finite type $A$-algebra $C$ and an isomorphism $B \cong C^\wedge $.
The rig-étale case of [III Theorem 7, Elkik]
Proof.
Choose a presentation $B = A[x_1, \ldots , x_ r]^\wedge /J$. By Lemma 88.8.2 part (6) we can find $c \geq 0$ and $f_1, \ldots , f_ r \in J$ such that $\det _{1 \leq i, j \leq r}(\partial f_ j/\partial x_ i)$ divides $a^ c$ in $B$ and $a^ c J \subset (f_1, \ldots , f_ r) + J^2$. Hence Lemma 88.7.2 applies. This finishes the proof, but we'd like to point out that in this case the use of Lemma 88.5.3 can be replaced by the much easier Lemma 88.5.5.
$\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)